The general relativistic infinite plane

Abstract

Uniform fields are one of the simplest and most pedagogically useful examples in introductory courses on electrostatics or Newtonian gravity. In general relativity there have been several proposals as to what constitutes a uniform field. In this article we examine two metrics that can be considered the general relativistic version of the infinite plane with finite mass per unit area. The first metric is the 4D version of the 5D "brane" world models which are the starting point for many current research papers. The second case is the cosmological domain wall metric. We examine to what extent these different metrics match or deviate from our Newtonian intuition about the gravitational field of an infinite plane. These solutions provide the beginning student in general relativity both computational practice and conceptual insight into Einstein's field equations. In addition they do this by introducing the student to material that is at the forefront of current research.Comment: Accepted for publication in the American Journal of Physic

Quasi-uniform gravitational field of a disk revisited

“…General properties of static plane-symmetric spacetimes have been investigated in Refs. [17,18,[23][24][25][26]. It should be noted that solution (5) describes a vacuum domain wall and there is a singularity at |z| = −1/K, K < 0.…”

“…The both metrics, (2) and (3), were much discussed (see Refs. [15,17,18] and references therein). We can also mention the important property proven in Refs.…”

“…The metrics of the uniform gravitational field presented in Refs. [11][12][13][14][15][16][17][18] are not obtained as a result of rigorous derivations of a metric defined by a known distribution of gravitational sources. To fulfill such a derivation, it is instructive to compare uniform and quasi-uniform fields in electrodynamics and gravity.…”

Compact and extended objects from self-interacting phantom fields

“…So we have solutions that tend asymptotically to the local minimum of the potential (5) While potentially unphysical, this negative energy density is not unexpected since in the case of some other types of planar solution [24], one also finds T − V. The subscript x indicates this is the pressure in the x direction. The different components of the scalar field energy-momentum tensor are calculated via (2).…”

“…Recall that (22) is redundant so in numerically solving the coupled system we only use (20), (21), (23), and (24). For a given value of χð0Þ there are only certain values of m 1 , m 2 which lead to regular solutions.…”